Symmetric function

Given any function ${\displaystyle f}$  in ${\displaystyle n}$  variables with values in an abelian group, a symmetric function can be constructed by summing values of ${\displaystyle f}$  over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ${\displaystyle f.}$  The only general case where ${\displaystyle f}$  can be recovered if both its symmetrization and antisymmetrization are known is when ${\displaystyle n=2}$  and the abelian group admits a division by 2 (inverse of doubling); then ${\displaystyle f}$  is equal to half the sum of its symmetrization and its antisymmetrization.

• Consider the real function $${\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).}$$  By definition, a symmetric function with ${\displaystyle n}$  variables has the property that $${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}}$$  In general, the function remains the same for every permutation of its variables. This means that, in this case, $${\displaystyle (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})}$$  and so on, for all permutations of ${\displaystyle x_{1},x_{2},x_{3}.}$ 
• Consider the function $${\displaystyle f(x,y)=x^{2}+y^{2}-r^{2}.}$$  If ${\displaystyle x}$  and ${\displaystyle y}$  are interchanged the function becomes $${\displaystyle f(y,x)=y^{2}+x^{2}-r^{2},}$$  which yields exactly the same results as the original ${\displaystyle f(x,y).}$ 
• Consider now the function $${\displaystyle f(x,y)=ax^{2}+by^{2}-r^{2}.}$$  If ${\displaystyle x}$  and ${\displaystyle y}$  are interchanged, the function becomes $${\displaystyle f(y,x)=ay^{2}+bx^{2}-r^{2}.}$$  This function is not the same as the original if ${\displaystyle a\neq b,}$  which makes it non-symmetric.

In statistics, an ${\displaystyle n}$ -sample statistic (a function in ${\displaystyle n}$  variables) that is obtained by bootstrapping symmetrization of a ${\displaystyle k}$ -sample statistic, yielding a symmetric function in ${\displaystyle n}$  variables, is called a U-statistic. Examples include the sample mean and sample variance.

• Alternating polynomial
• Elementary symmetric polynomial – Mathematical function
• Even and odd functions – Functions such that f(–x) equals f(x) or –f(x)
• Exchangeable random variables – Concept in statistics
• Quasisymmetric function
• Ring of symmetric functions
• Symmetrization – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback
• Vandermonde polynomial – determinant of Vandermonde matrixPages displaying wikidata descriptions as a fallback

• F. N. David, M. G. Kendall & D. E. Barton (1966) Symmetric Function and Allied Tables, Cambridge University Press.
• Joseph P. S. Kung, Gian-Carlo Rota, & Catherine H. Yan (2009) Combinatorics: The Rota Way, §5.1 Symmetric functions, pp 222–5, Cambridge University Press, ISBN 978-0-521-73794-4.